4 351 1st order system, part 2

4 351 1st order system, part 2 - 1st Order Systems response...

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1 1 1 st Order Systems response to a sine function input 2 1st-Order Systems yyK F ( t ) τ += ± For systems with storage/dissipative elements but negligible inertia : where dy y dt = ± and τ is the time constant of the system, a measure of the speed of the system response.

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2 3 1st-Order System: Sine Input yyK F ( t )K A s i nt τ ω + == ± F(t) = KAsin ω t, y(0) = y 0 Again, the solution is found by adding () 1 2 1 t/ KA y( t ) Ce sin t tan ωτ −− =+ + where C depends on y 0 . steady state response transient response Same as before. Usually not of interest when looking at the steady state response. Æ Ignore for the remaining analysis. (Don’t forget it’s there, though, at short times.) hp y(t )y ( t ( t ) = + The solution, which can be verified by substitution, is 4 1st-Order System: Sine Input 1 2 1 KA y( t ) Ce sin t tan + B( ) φ ( ) ( ) y(t ) Ce B sin t ωφω + • The output has the same frequency as the input. F ( t A s i + ± • The amplitude is proportional to the input amplitude , but modified by a frequency-dependent term.
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4 351 1st order system, part 2 - 1st Order Systems response...

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